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Machines, Work, and Energy
The Egyptian pyramids were built about 4,000 years
ago. It took workers approximately 80 years to build
the pyramids at Giza. The largest, called the Great
Pyramid, contains about 1 million stone blocks, each
weighing about 2.5 tons. How were the ancient
Egyptians able to build such an incredible monument?
What did the ancient Egyptians use to help them build
the pyramids? Egyptologists, men and women that
study ancient Egypt, disagree about the details of how
the gigantic structures were built, but most agree that
a system of ramps and levers (simple machines) was
necessary for moving and placing the blocks. The fact
that they could move such enormously massive blocks
of limestone to build the Great Pyramid to a height of
481 feet (roughly equivalent to a 48-story building) is
fascinating, don't you think? Perhaps the most
amazing part of this story is that the Great Pyramid at
Giza still stands, and is visited by tens of thousands of
people each year.
Key Questions
3 Why does stretching a rubber
band increase its potential
energy?
3 How much power can a highly
trained athlete have?
3 What is one of the most perfect
machines ever invented?
3 Why does time always move
forward, and never backward?
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4.1 Work and Power
Energy is a measure of an object’s ability to do work. Suppose you lift your book over your head.
The book gets potential energy which comes from your action. Now suppose you lift your book
fast, then lift it again slowly. The energy is the same because the height is the same. But it feels
different to transfer the energy fast or slow. The difference between moving energy fast or slow is
described by power. Power is the rate at which energy flows or at which work is done. This section
is about power and its relation to work and energy.
Reviewing the definition of work
What “work” In the last chapter you learned that work has a very specific meaning in
means in physics. Work is the transfer of energy that results from applying a force over
physics a distance. If you push a block with a force of one newton for a distance of
Vocabulary
power, watt, horsepower
Objectives
3 Define work in terms of force and
distance and in terms of energy.
3 Calculate the work done when
moving an object.
3 Explain the relationship between
work and power.
one meter, you do one joule of work. Both work and energy are measured in
the same units (joules) because work is a form of energy.
Work is done When thinking about work you should always be clear about which force is
on objects doing the work. Work is done by forces on objects. If you push a block one
meter with a force of one newton, you have done one joule of work
(Figure 4.1).
Energy An object that has energy is able to do work; without energy, it is impossible
is needed to to do work. A block that slides across a table has kinetic energy that can be
do work used to do work. If the block hits a ball, it will do work on the ball and change
its motion. Some of the block’s kinetic energy is transferred to the ball. An
elastic collision is a common method of doing work.
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4.1 WORK AND POWER
Figure 4.1: A force of 1 newton applied for
1 meter does one joule of work on the block.
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CHAPTER 4: MACHINES, WORK, AND ENERGY
Work and energy
Work and Doing work always means transferring energy. The energy may be transferred
potential energy to the object you apply the force to, or it may go somewhere else. You can
increase the potential energy of a rubber band by exerting a force that stretches
it. The work you do stretching the rubber band is stored as energy of the rubber
band. The rubber band can then use the energy to do work on a paper airplane,
giving it kinetic energy (Figure 4.2).
Work may not
increase the
energy of an
object
You can do work on a block by sliding it across a level table. In this example,
though, the work you do does not increase the energy of the block. Because the
block will not slide back all by itself, it does not gain the ability to do work
itself, therefore gains no energy. Your work is done to overcome friction. The
block does gain a tiny bit of energy because its temperature rises slightly from
friction. However, that energy comes from the force of friction, not from your
applied force.
Not all force Sometimes force is applied to an object, but no work is done. If you push
does work down on a block sitting on a table and it doesn’t move, you have not done any
work on the block (force A below). If you use W=Fd to calculate the work,
you will get zero no matter how strong the force because the distance is zero.
Force at an angle There are times when only some of a force does work. Force B is applied at an
to distance angle to the direction of motion of a block. Only a portion of the force is in the
direction the block moves, so only that portion of the force does work.
Doing The more exact calculation of work
the most work is the product of the portions of force
and distance that are in the same
direction. To do the greatest amount
of work, you must apply force in the
direction the object will move (force
C). If forces A, B, and C have equal
strengths, force C will do the most
work because it is entirely in the
direction of the motion.
Figure 4.2: You can do work to increase an
object’s potential energy.Then the potential
energy can be converted to kinetic energy.
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Work done against gravity
Lifting force Many situations involve work done by or against the force of gravity. To lift
equals the something off the floor, you must apply an upward force with a strength equal
weight to the object’s weight. The work done while lifting an object is equal to its
change in potential energy. It does not matter whether you lift the object
straight up or you carry it up the stairs in a zigzag pattern. The work is the
same in either case. Work done against gravity is calculated by multiplying
the object’s weight by its change in height.
Why the path The reason the path does not matter is found in the definition of work as the
does not matter force times the distance moved in the direction of the force. If you move an
object on a diagonal, only the vertical distance matters because the force of
gravity is vertical (Figure 4.3). It is much easier to climb stairs or go up a
ramp but the work done against gravity is the same as if you jumped straight
up. Stairs and ramps are easier because you need less force. But you have to
apply the force over a longer distance. In the end, the total work done against
gravity is the same no matter what path you take.
Figure 4.3: The work done when lifting an
object equals its mass multiplied by the
strength of gravity multiplied by the change in
height.
Alexander has a mass of 70 kilograms. His apartment is on the second floor, 5 meters up from ground level. How much work
does he do against gravity each time he climbs the stairs to his apartment?
Calculating
work
1. Looking for:
You are asked for the work.
2. Given:
You are given the mass in kilograms and the height in meters.
3. Relationships:
Fg=mg
4. Solution:
The force is equal to Alexander’s weight.
Fg = (70 kg)(9.8 m/sec2) Fg = 686 N
W=Fd
Use the force to calculate the work.
W = (686 N)(5 m) W = 3430 J
He does 3430 joules of work.
Your turn...
a. How much additional work does Alexander have to do if he is carrying 5 kilograms of groceries? Answer: 245 J
b. A car engine does 50,000 J of work to accelerate at 10 m/sec2 for 5 meters. What is the mass of the car? Answer: 1,000 kg
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Power
What is power? Suppose Michael and Jim each lift a barbell weighing 100 newtons from the
ground to a height of two meters (Figure 4.4). Michael lifts quickly and Jim
lifts slowly. Because the barbell is raised the same distance, it gains the same
amount of potential energy in each case. Michael and Jim do the same amount
of work. However, Michael’s power is greater because he gets the work done
in less time. Power is the rate at which work is done.
Units of power The unit for power is equal to the unit of work (joules) divided by the unit of
time (seconds). One watt is equal to one joule per second. The watt was
named after James Watt (1736-1819), the Scottish engineer who invented the
steam engine. Another unit of power that is often used for engine power is the
horsepower. Watt expressed the power of his engines as the number of
horses an engine could replace. One horsepower is equal to 746 watts.
Calculating work So how much power do Michael and Jim use? You must first calculate the
Figure 4.4: Michael and Jim do the same
amount of work but do not have the same
power.
work they do, using W = Fd. The force needed to lift the barbell is equal to its
weight (100 N). The work is therefore 100 newtons times two meters, or 200
joules. Each of them does 200 joules of work.
Calculating To find Michael’s power, divide his work (200 joules) by his time (1 second).
power Michael has a power of 200 watts. To find Jim’s power, divide his work (200
joules) by his time (10 seconds). Jim’s power is 20 watts. Jim takes 10 times as
long to lift the barbell, so his power is one-tenth as great.
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Calculating power
Human power The maximum power output of a person is typically around a few hundred
watts. However, it is only possible to keep up this power for a short time.
Highly trained athletes can keep up a power of 350 watts for about an hour.
An average person running or biking for a full hour produces an average
power of around 200 watts.
A roller coaster is pulled up a hill by a chain attached to a motor. The roller
coaster has a total mass of 10,000 kg. If it takes 20 seconds to pull the roller
coaster up a 50-meter hill, how powerful is the motor?
Calculating
power
1. Looking for:
You are asked for power.
2. Given:
You are given the mass in kilograms, the time in
seconds, and the height in meters.
3. Relationships:
Fg=mg
4. Solution:
Calculate the weight of the roller coaster:
Fg = (10,000 kg)(9.8 m/sec2) Fg = 98,000 N
Calculate the work:
W = (98,000 N)(50 m) W = 4,900,000 J or 4.9 ×106 J
Calculate the power:
P = (4.9 × 106 J) ÷ (20 sec) P = 245,000 watts
W=Fd
P=W/t
Your turn...
a. What would the motor’s power be if it took 40 seconds to pull the same roller coaster up the hill? Answer: 122,500 watts
b. What is the power of a 70-kilogram person who climbs a 10-meter-high hill in 45 seconds? Answer: 152 watts
4.1 Section Review
1.
2.
3.
4.
Explain how work is related to energy.
Who does more work, a person who lifts a 2-kilogram object 5 meters or a person who lifts a 3-kilogram object 4 meters?
While sitting in class, your body exerts a force of 600 N on your chair. How much work do you do?
Is your power greater when you run or walk up a flight of stairs? Why?
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CHAPTER 4: MACHINES, WORK, AND ENERGY
4.2 Simple machines
Vocabulary
How do you move something that is too heavy to carry? How did the ancient Egyptians build the
pyramids long before the invention of powered machines? The answer to these questions has to do
with the use of simple machines. In this section, you will learn how simple machines multiply
forces to accomplish many tasks.
machine, input, output, fulcrum,
simple machines, mechanical
advantage, fulcrum, input arm,
output arm, tension
Using machines
What technology Today’s technology allows us to do incredible things. Moving huge steel
allows us to do beams, digging tunnels that connect two islands, and building 1,000-foot
skyscrapers are examples. What makes these accomplishments possible? Have
we developed super powers since the days of our ancestors?
What is In a way we have developed super powers. Our powers come from the clever
a machine? human invention of machines. A machine is a device with moving parts that
Objectives
3 Describe how a machine works in
terms of input and output.
3 Define simple machines and
name some examples.
3 Calculate the mechanical
advantage of a simple machine
given the input and output force.
work together to accomplish a task. A bicycle is made of a combination of
machines that work together. All the parts of a bicycle work as a system to
transform forces from your muscles into motion. A bicycle allows you to
travel at faster speeds and for greater distances than possible on foot.
The concepts of Machines are designed to do something. To understand how machines work it
input and output is useful to define an input and an output. The input includes everything you
do to make the machine work, like pushing on the bicycle pedals, for instance.
The output is what the machine does for you, like going fast or climbing a
steep hill. For the machines that are the subject of this chapter, the input and
output may be force, power, or energy.
Figure 4.5: A bicycle contains machines
working together.
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Simple machines
The beginning of The development of the technology that created cars, airplanes, and other
technology modern conveniences began with the invention of simple machines. A
simple machine is an unpowered mechanical device that accomplishes a task
with only one movement (such as a lever). A lever allows you to move a rock
that weighs 10 times (or more) what you weigh. Some important types of
simple machines are shown below.
Figure 4.6: A small input force can create
a large output force if the lever is arranged
correctly.
Input force and Simple machines work with forces. The input force is the force you apply to
output force the machine. The output force is the force the machine applies to what you
are trying to move. Figure 4.6 shows how a lever can be arranged to create a
large output force from a small input force.
Ropes and A rope and pulley system is a simple machine made by connecting a rope to
pulleys one or more pulleys. You apply the input force to the rope and the output
force is exerted on the load you are lifting. One person could easily lift an
elephant with a properly designed system of pulleys (Figure 4.7).
Machines within Most of the machines we use today are made up of combinations of different
machines types of simple machines. For example, the bicycle uses wheels and axles,
levers (the pedals and kickstand), and gears. If you take apart a complex
machine such as a video cassette recorder, a clock, or a car engine, you will
find many simple machines inside. In fact, a VCR contains simple machines
of every type including screws, ramps, pulleys, wheels, gears, and levers.
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4.2 SIMPLE MACHINES
Figure 4.7: A simple machine made with a
rope and pulley allows one person to lift
tremendous loads.
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Mechanical advantage
Ratio of output Simple machines are best understood through the concepts of input and output
to input force forces. The mechanical advantage of a machine is the ratio of the output
force to the input force. If the mechanical advantage of a machine is larger
than one, the output force is larger than the input force. A mechanical
advantage smaller than one means the output force is smaller than the input
force. Mechanical advantage is a ratio of forces, so it is a pure number without
any units.
What is the mechanical advantage of a lever that allows Jorge to lift a 24-newton
box with a force of 4 newtons?
Calculating
mechanical
advantage
1. Looking for:
You are asked for the mechanical advantage.
2. Given:
You are given the input force and the output force in newtons.
3. Relationships:
MA=Fo/Fi
4. Solution:
MA = (24 N)/(4 N)
MA = 6
a. Calculate the mechanical advantage of a rope and pulley system that requires 10 newtons of force to lift a 200-newton load.
Answer: 20
b. You use a block and tackle with a mechanical advantage of 30. How heavy a load can you lift with an input force of 100 N?
Answer: 3000 N
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Work and machines
Input and output A simple machine does work because it exerts forces over a distance. If you
work are using the machine you also do work, because you apply forces to the
machine that move its parts. By definition, a simple machine has no source of
energy except the immediate forces you apply. That means the only way to
get output work from a simple machine is to do input work on the machine. In
fact, the output work done by a simple machine can never exceed the input
work done on the machine. This is an important result.
Perfect In a perfect machine the output work equals the input work. Of course, there
machines are no perfect machines. Friction always converts some of the input work to
heat and wear, so the output work is always less than the input work.
However, for a well-designed machine, friction can be small and we can often
assume input and output work are approximately equal.
An example Figure 4.8 shows a simple machine that has a mechanical advantage of two.
The machine lifts a 10-newton weight a distance of one-half meter. The
output work is five joules (10 N × 0.5 m).
You must do at least five joules of work on the machine to lift the weight. If
you assume the machine is perfect, then you must do exactly 5 J of input
work to get 5 J of output work. The input force is only five newtons since the
machine has a mechanical advantage of two. That means the input distance
must be 1 meter because 5 N × 1 m = 5 J. You have to pull one meter of rope
to raise the weight one-half meter.
The cost of The output work of a machine can never be greater than the input work. This
multiplying force is a rule that is true for all machines. Nature does not give something for
nothing. When you design a machine that multiplies force, you pay by having
to apply the force over a greater distance.
The force and distance are related by the amount of work done. In a perfect
(theoretical) machine, the output work is exactly equal to the input work. If
the machine has a mechanical advantage greater than one, the input force is
less than the output force. However, the input force must be applied over a
longer distance to satisfy the rule about input and output work.
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4.2 SIMPLE MACHINES
Figure 4.8: The work output equals the
work input even though the forces differ.
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Using work to solve problems
Mechanical To solve mechanical advantage problems, start by assuming a perfect machine,
advantage with nothing lost to friction. Set the input and output work equal and use this
relationship to find the mechanical advantage.
Force or Many problems give three of the four quantities: input force, input distance,
distance output force, and output distance. If the input and output work are equal then
force × distance at the input of the machine equals force × distance at the
output. Using this equation, you can solve for the unknown force or distance.
Work and
machines
A jack is used to lift one side of a car in order to replace a tire. To lift the car, the jack handle moves
30 centimeters for every one centimeter that the car is lifted. If a force of 150 newtons is applied to
the jack handle, what force is applied to the car by the jack? You can assume all of the input work
equals output work.
1. Looking for:
You are asked for the output force in newtons.
2. Given:
You are given the input force in newtons, and the input distance and output
distance in centimeters. Convert these distances to meters.
3. Relationships:
Work =Force × distance
4. Solution:
Input work: W = (150 N)(0.30 m) = 45 joules
Output work: 45 J of input work = Force × 0.01 m
F = 45 J/ 0.01 m = 4,500 newtons
The jack exerts an upward force of 4,500 newtons on the car.
and
Input work = output work
,
Your turn...
a. A mover uses a pulley to lift a 2,400-newton piano up to the second floor. Each time he pulls
the rope down 2 meters (input distance), the piano moves up 0.25 meter (output distance).
With what force does the mover pull on the rope? Answer: 300 newtons
b. A nutcracker is a very useful lever. The center of the nutcracker (where the nut is) moves
one centimeter for each two centimeters your hand squeezes down. If a force of 40 newtons
is needed to crack the shell of a walnut, what force must you apply? Answer: 20 newtons
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How a lever works
Example A lever can be made by balancing a board on a log (Figure 4.9). Pushing
of a lever down on one end of the board lifts a load on the other end of the board. The
downward force you apply is the input force. The upward force the board
exerts on the load is the output force.
Parts of All levers include a stiff structure that rotates around a fixed point called the
the lever fulcrum. The side of the lever where the input force is applied is called the
input arm. The output arm is the end of the lever that applies the output
force. Levers are useful because you can arrange the fulcrum and the input
and output arms to make almost any mechanical advantage you need.
Changing When the fulcrum is in the middle of the lever, the input and output forces are
direction the same. An input force of 100 newtons makes an output force of 100
newtons. The input and output forces are different if the fulcrum is not in the
center of the lever (Figure 4.10). The side of the lever with the longer arm has
the smaller force. If the input arm is 10 times longer than the output arm, the
output force is 10 times greater than the input force.
Figure 4.9: A board and log can make a
lever used to lift a rock.
Mechanical You can find the mechanical advantage of a lever by looking at two triangles.
advantage The output work is the output force multiplied by the output distance. The
of a lever input work is the input distance multiplied by the input force. By setting the
input and output work equal, you see that the ratio of forces is the inverse of
the ratio of distances. The larger (input) distance has the smaller force. The
ratio of distances is equal to the ratio of the lengths of the two arms of the
lever. Using the lengths of the arms is the easiest way to calculate the
mechanical advantage of a lever (below).
Figure 4.10: How to determine the
mechanical advantage of a lever.
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Types of levers
The output force You can also make a lever in which the output force is less than the input
can be less than force. The input arm is shorter than the output arm on this kind of lever. You
the input force might design a lever this way if you need the motion on the output side to be
larger than the motion on the input side. A very small downward motion on the
input side can cause the load to lift a large distance on the output side.
The three types Levers are used in many common machines, including, for example, pliers, a
of levers wheelbarrow, and the human biceps and forearm (Figure 4.11). You may have
heard the human body described as a machine. In fact, it is a machine. Bones
and muscles work as levers when you do something as simple as biting an
apple. Levers are classified as one of three types or classes defined by the
location of the input and output forces relative to the fulcrum. The mechanical
advantage is always the ratio of lengths of the input arm to the output arm.
A lever has a mechanical advantage of 4. Its input arm is 60 centimeters long. How
long is its output arm?
Mechanical
advantage
of levers
1. Looking for:
You are asked for the output arm in centimeters.
2. Given:
You are given the mechanical advantage and the length of the
input arm in centimeters.
3. Relationships:
MAlever =
4. Solution:
4=
60 cm
Lo
Li
Lo
4 Lo = 60 cm
Lo =
Figure 4.11: There are three classes of
60 cm
= 15 cm
4
levers.
a. What is the mechanical advantage of a lever with an input arm of 25 centimeters
and an output arm of 100 centimeters? Answer: 0.25
b. A lever has an input arm of 100 centimeters and an output arm of 10 centimeters.
What is the mechanical advantage of this lever? Given this mechanical
advantage, how much input force is needed to lift a 100-newton load?
Answer: MAlever = 10; 10 newtons of force would be needed.
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How a rope and pulley system works
Tension in ropes Ropes and strings carry forces along their length. The force in a rope is called
and strings tension and is a pulling force that acts along the direction of the rope. The
tension is the same at every point in a rope. If the rope is not moving, its
tension is equal to the force pulling on each end (below). Ropes or strings do
not carry pushing forces. This is obvious if you ever tried pushing a rope.
The forces in Figure 4.12 shows three different configurations of ropes and pulleys.
a pulley system Imagine pulling with an input force of 5 newtons. In case A, the load feels a
force equal to your input force. In case B, there are two strands of rope
supporting the load, so the load feels twice your input force. In case C, there
are three strands so the output force is three times the input force.
Mechanical The mechanical advantage of a pulley system depends on the number of
advantage strands of rope directly supporting the load. In case C, three strands directly
support the load, so the output force is three times the input force. The
mechanical advantage is 3. To make a rope and pulley system with a greater
mechanical advantage, you can increase the number of strands directly
supporting the load by taking more turns around the pulleys.
Work To raise the load 1 meter in case C, the input end of the rope must be pulled
for 3 meters. This is because each of the three supporting strands must
shorten by 1 meter. The mechanical advantage is 3 but the input force must be
applied for three times the distance as the output force. This is another
example of the rule stating that output and input work are equal for a perfect
machine.
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4.2 SIMPLE MACHINES
Figure 4.12: A rope and pulley system can
be arranged to have different mechanical
advantages.
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Gears and ramps
Rotating motion Many machines require that rotating motion be transmitted from one place to
another. The transmission of rotating motion is often done with gears
(Figure 4.13). Some machines that use gears, such as small drills, require
small forces at high speeds. Other machines, such as the paddle wheel on the
back of a steamboat, require large forces at low speed.
How gears work The rule for how two gears turn depends on the number of teeth on each gear.
The teeth don’t slip, so moving 36 teeth on one gear means that 36 teeth have
to move on any connected gear. Suppose a large gear with 36 teeth is
connected to a small gear with 12 teeth. As the large gear turns once around, it
moves 36 teeth on the smaller gear. The smaller gear must turn three times
(3 × 12 = 36) for every single turn of the large gear (Figure 4.13).
Ramps A ramp is another type of simple machine. Using a ramp allows you to push a
heavy car to a higher location with less force than is needed to lift the car
straight up. Ramps reduce the input force needed by increasing the distance
over which the input force acts. For example, suppose a 10-meter ramp is used
to lift a car one meter. The output work is work done against gravity. If the
weight of the car is 500 newtons, then the output work is 500 joules
(w = mgh = 500 N × 1 m).
Figure 4.13: The smaller gear makes three
turns for each one turn of the larger gear.
Figure 4.14: The car must be pulled
Mechanical The input work is the input force multiplied by the length of the ramp (10
advantage of a meters). If you set the input work equal to the output work, you quickly find
ramp that the input force is 50 newtons (Fd = F × 10 m = 500 J). The input force is
10 meters to lift it up one meter, but only
one-tenth the force is needed
one-tenth of the output force. For a frictionless ramp, the mechanical
advantage is the length of the ramp divided by the height (Figure 4.14).
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Screws
Screws A screw is a simple machine that turns rotating motion into linear motion
(Figure 4.15). A screw works just like a ramp that curves as it gets higher.
The “ramp” on a screw is called a thread. Imagine unwrapping one turn of a
thread to make a straight ramp. Each turn of the screw advances the nut the
same distance it would have gone sliding up the ramp. The lead of a screw is
the distance it advances in one turn. A screw with a lead of one millimeter
advances one millimeter for each turn.
A screw and The combination of a screw and screwdriver has a very large mechanical
screwdriver advantage. The mechanical advantage of a screw is found by thinking about it
as a ramp. The vertical distance is the lead of the screw. The length of the
ramp is measured as the average circumference of the thread. A quarter-inch
screw in a hardware store has a lead of 1.2 millimeters and a circumference of
17 millimeters along the thread. The mechanical advantage is 14. If you use a
typical screwdriver with a mechanical advantage of 4, the total mechanical
advantage is 14 × 4 or 56 (theoretically). Friction between the screw and the
mating surface causes the actual mechanical advantage to be somewhat less
than the theoretical value, but still very large (Figure 4.16).
Figure 4.15: A screw is a rotating ramp.
4.2 Section Review
1. Name two simple machines that are found on a bicycle.
2. Calculate the mechanical advantage of the crowbar
shown at right.
3. Classify each of these as a first-, second-, or thirdclass levers: see-saw, baseball bat, door on hinges,
scissors (Figure 4.16).
4. A large gear with 48 teeth is connected to a small
gear with 12 teeth. If the large gear turns twice, how
many times must the small gear turn?
5. What is the mechanical advantage of a 15 meter ramp that rises 3 meters?
Figure 4.16: Which type of lever is shown
in each picture?
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CHAPTER 4: MACHINES, WORK, AND ENERGY
4.3 Efficiency
Vocabulary
So far we have talked about perfect machines. In a perfect machine there is no friction and the
output work equals the input work. Of course, there are no perfect machines in human technology.
This section is about efficiency, which is how we measure how close to perfect a machine is. The
bicycle comes as close to perfect as any machine ever invented. Up to 95 percent of the work done
by the rider on the pedals becomes kinetic energy of the bicycle (Figure 4.17). Most machines are
much less perfect. An automobile engine converts less than 15 percent of the chemical energy in
gasoline into output work to move a car.
Friction
Friction Friction is a catch-all term for many processes that oppose motion. Friction
can be caused by rubbing or sliding surfaces. Friction can also be caused by
moving through liquid, such as oil or water. Friction can even be caused by
moving though air, as you can easily feel by sticking your hand out the
window of a moving car.
efficiency, reversible, irreversible
Objectives
3 Describe the relationship
between work and energy in a
simple machine.
3 Use energy conservation to
calculate input or output force
or distance.
3 Explain why a machine’s input
and output work can differ.
Friction and Friction converts energy of motion to heat and wear. The brakes on a car use
energy friction to slow the car down and they get hot. Over time, the material of the
brakes wears away. This also takes energy because the bonds between atoms
are being broken as material is being worn down. When we loosely say that
energy is “lost” to friction, the statement is not accurate. The energy is not lost,
but converted to other forms of energy that are difficult to recover and reuse.
Machines In an actual machine, the output work is less than the input work because of
friction. When analyzing a machine it helps to think like the diagram below.
The input work is divided between output work and “losses” due to friction.
Figure 4.17: A bicycle is highly efficient.
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Efficiency
100 percent A machine has an efficiency of 100 percent if the work output of the
efficient machine is equal to the work input. If a machine is 100 percent efficient, no
energy is diverted by friction or other factors. Although it is impossible to
create a machine with 100 percent efficiency, people who design machines
try to achieve as high an efficiency as possible.
The definition of The efficiency of a machine is the ratio of work output to work input.
efficiency Efficiency is usually expressed in percent. A machine that is 75 percent
efficient can produce three joules of output work for every four joules of
input work (Figure 4.18). One joule out of every four (25 percent) is lost to
friction. You calculate efficiency by dividing the work output by the work
input. You can convert the ratio into a percent by multiplying by 100.
Improving An important way to increase the efficiency of a machine is to reduce friction.
efficiency Ball bearings and oil reduce rolling friction. Slippery materials such as
TeflonTM reduce sliding friction. Designing a car with a streamlined shape
reduces air friction. All these techniques increase efficiency.
Figure 4.18: If the input work is four
joules, and the output work is three joules,
then the efficiency is 75 percent.
A person uses a 75-newton force to push a 51-kilogram car up a ramp. The ramp is 10 meters long and rises one meter.
Calculate the efficiency.
Calculating
efficiency
1. Looking for:
You are asked for the efficiency.
2. Given:
You are given the input force and distance, and the mass and height for the output.
3. Relationships:
Efficiency = Output work / Input work. Input: W = FD. Output: work done against gravity (W = mgh)
4. Solution:
Output work = (51 kg)(9.8 N/kg)(1 m) = 500 joules
Input work = (75 N)(10 m) = 750 joules
Efficiency = 500 J ÷ 750 J = 67%
Your turn...
a. If a machine is 80 percent efficient, how much input work is required to do 100 joules of output work?. Answer: 125 J
b. A solar cell needs 750 J of input energy to produce 100 J of output. What is its efficiency? Answer: 13.3%
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4.3 EFFICIENCY
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CHAPTER 4: MACHINES, WORK, AND ENERGY
Efficiency and time
A connection The efficiency is less than 100 percent for virtually all processes that convert
energy to any other form except heat. Scientists believe this is connected to
why time flows forward and not backward. Think of time as an arrow pointing
from the past into the future. All processes move in the direction of the arrow,
never backward (Figure 4.19).
Reversible Suppose a process were 100 percent efficient. As an example, think about
processes connecting two marbles of equal mass by a string passing over an ideal pulley
with no mass and no friction (Figure 4.20). One marble can go down,
transferring its potential energy to the other marble, which goes up. The
motion of the marble is reversible because it can go forward and backward as
many times as you want. In fact, if you watched a movie of the marbles
moving, you could not tell if the movie were playing forward or backward.
Figure 4.19: Time can be thought of as an
arrow pointing from the past into the future.
Friction and Now suppose there is a tiny amount of friction so the efficiency is 99 percent.
the arrow of time Because some potential energy is lost to friction, every time the marbles
exchange energy, some is lost and the marbles don’t rise quite as high as they
did the last time. If you made a movie of the motion, you could tell whether
the movie was running forward or backward. Any process with an efficiency
less than 100 percent runs only one way, forward with the arrow of time.
Irreversible Friction turns energy of motion into heat. Once energy is transformed into
processes heat, the energy cannot ever completely get back into its original form.
Because heat energy cannot get back to potential or kinetic energy, any process
with less than 100% efficiency is irreversible. Irreversible processes can only
go forward in time. Since processes in the universe almost always lose a little
energy to friction, time cannot run backward.
Figure 4.20: Exchanging energy with a
perfect, frictionless, massless pulley.
4.3 Section Review
1.
2.
3.
4.
What is the relationship between work and energy in a machine?
Why can the output work of a simple machine never be greater than the input work?
Use the concept of work to explain the relationship between input and output forces and distances.
How does the efficiency of a car compare to the efficiency of a bicycle? Why do you think there is such a large difference?
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Prosthetic Legs and Technology
The human leg is a complex and versatile machine. Designing a
prosthetic (artificial) device to match the leg’s capabilities is a
serious challenge. Teams of scientists, engineers, and designers
around the world use different approaches and technologies to
develop prosthetic legs that help the user regain a normal, active
lifestyle.
Studying the human gait cycle
Each person has a unique way of walking. But studying the way
humans walk has revealed that some basic mechanics hold true for
just about everyone. Scientists analyze how we walk by looking at
our “gait cycle.” The gait cycle consists of two consecutive strides
while walking, one foot and then the other. By breaking the cycle
down into phases and figuring out where in the sequence
prosthetics devices could be improved, designers have added
features and materials that let users walk safely and comfortably
with their own natural gait.
Designing a better prosthetic leg
In many prosthetic leg designs, the knee is the component that
controls how the device operates. In the past, most designs were
basic and relied on the user learning how to walk properly. This
effort required up to 80% more energy than a normal gait and often
made walking with an older prosthetic leg a work out!
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The knee joint in those older designs was
often a hinge that let the lower leg swing
back and forth. The hinge could also lock in
place to keep the leg straight and support
the user's weight to make standing easier.
This type of system worked relatively
well on level surfaces, but could be
difficult to use on inclines, stairs,
slightly irregular terrain (like a hiking
trail), or slippery surfaces.
Current prosthetic legs have improved
upon old designs by employing
hydraulics, carbon fiber, mechanical
linkages, motors, computer
microprocessors, and innovative
combinations of these technologies to
give more control to the user. For
example, in some designs a device called
a damper helps to control how fast the
lower leg can swing back and forth while
walking. The damper accomplishes this
by changing the knee’s resistance to
movement as needed.
New knee designs allow users to walk, jog,
and with some models even run with a
more natural gait. In fact, in 2003 Marlon
Shirley became the first above-the-knee
amputee in the world to break the 11-second
barrier in the 100-meter dash with a time
of 10.97 seconds! She accomplished
this feat with the aid of a special
prosthetic leg designed
specifically for sprinting.
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Designs that learn
By continuously monitoring the velocities of
the upper and lower leg, the angle of knee
bend, changes in the terrain, and other data,
computer microprocessors in the knee calculate
and make adjustments to changing conditions in
milliseconds. This makes the prosthetic leg
more stable and efficient, allowing the knee,
ankle, and foot to work together as a unit. Some
designs have built-in memory systems that store
information from sensors about how the user
walks. These designs “learn” how to make finetuned adjustments based on the user’s particular
gait pattern.
New foot designs
New foot designs also reduce the energy required
to walk with prosthetic leg systems. They also
smooth out the user’s stride. Using composite
materials, these designs allow the foot to flex in
different ways during the gait cycle. Both the heel
and the front part of the foot act like springs to
store and then release energy. When the foot first
strikes the ground, the heel flexes and absorbs
some of the energy, reducing the impact.
Weight gets shifted toward the front
of the foot as the walker moves
through the stride.
As this happens, the heel springs back into shape and the energy
released helps to flex the front part of the foot, once again storing
energy. When the foot leaves the ground in the next part of the gait
cycle, the flexed front part of the foot releases its' stored energy and
helps to push the foot forward into the next stride.
Designers have realized the advantage of making highly specialized
feet that match and sometimes exceed the capabilities of human
feet. Distance running and sprinting feet are built to different
specifications to efficiently deal with the forces and demands
related to these activities.
A rock-climbing inventor
Hugh Herr, Ph.D., a physicists and engineer at the Harvard-MIT
Division of Health Sciences and Technology (Boston,
Massachusetts), studies biomechanics and prosthetic technology. In
addition to holding several patents in this field, he has developed
highly specialized feet for rock climbing that are small and thin—
ideal for providing support on small ledges. Being an accomplished
climber and an amputee allows Herr to field test his own
inventions. While rock climbing, he gains important insights into
the effectiveness and durability of each design.
Questions:
1. What are some technologies used by designers of prosthetic
legs to improve their designs?
2. How are computers used to improve the function of prosthetic
devices?
3. Explain how new foot designs reduce the amount of energy
required to walk with a prosthetic leg.
4. Research the field of biomechanics. In a paragraph:
(1) describe what the term “biomechanics” means, and
(2) write about a biomechanics topic that interests you.
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Chapter 4 Review
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Understanding Vocabulary
Reviewing Concepts
Select the correct term to complete the sentences.
Section 4.1
mechanical advantage
machine
irreversible
watt
tension
input arm
horsepower
input
work
fulcrum
output
power
simple machines
efficiency
Section 4.1
1.
A unit of power equal to 746 watts is called one ____.
2.
____ is the rate of doing work.
3.
Force multiplied by distance is equal to ____.
4.
The measurement unit of power equal to one joule of work performed
in 1 second is called the ____.
1.
Why are work and energy both measured in joules?
2.
If you lift a box of books one meter off the ground, you are doing
work. How much more work do you do by lifting the box 2 meters off
the ground?
3.
Decide whether work is being done (using your physics definition of
work) in the following situations:
a. Picking up a bowling ball off the floor.
b. Two people pulling with the same amount of force on each end
of a rope.
c. Hitting a tennis ball with a tennis racket.
d. Pushing hard against a wall for an hour.
e. Pushing against a book so it slides across the floor.
f. Standing very still with a book balanced on your head.
Section 4.2
5.
The ramp, the lever, and the wheel and axle are examples of ____.
4.
6.
Pushing on the pedals of a bicycle is an example of the ____ to a
machine.
In which direction should you apply a force if you want to do the
greatest amount of work?
5.
What is the difference between work and power?
7.
Moving a heavy load is an example of the ____ from a lever.
6.
What is the meaning of the unit of power called a watt?
8.
To calculate a machine’s ____, you divide the output force by the
input force.
Section 4.2
9.
A ____ is a device with moving parts that work together to
accomplish a task.
7.
List five types of simple machines.
8.
Which two types of simple machines are in a wheelbarrow?
9.
A certain lever has a mechanical advantage of 2. How does the lever’s
output force compare to the input force?.
10. The pivot point of a lever is known as its ____.
11. The side of a lever where the input force is applied is the _____.
12. The pulling force in a rope is known as ____.
Section 4.3
13. ____ is the ratio of work output to work input and is usually
expressed as a percent.
14. A process with less than 100% efficiency is _____.
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CHAPTER 4: MACHINES, WORK, AND ENERGY
10. Can simple machines multiply input forces to get increased output
forces? Can they multiply work input to increase the work output?
11. Draw a diagram of each of the three types of levers. Label the input
force, output force, and fulcrum on each.
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CHAPTER 4 REVIEW
12. You and a friend pull on opposite ends of a rope. You each pull with a
force of 10 newtons. What is the tension in the rope?
6.
Two cranes use rope and pulley systems to lift a load from a truck to
the top of a building. Crane A has twice as much power as crane B.
a. If it takes crane A 10 seconds to lift a certain load, how much
time does crane B take to lift the same load?
b. If crane B can do 10,000 joules of work in a minute, how many
joules of work can crane A do in a minute?
7.
An elevator lifts a 500 kg load up a distance of 10 meters in 8
seconds.
a. Calculate the work done by the elevator.
b. Calculate the elevator’s power.
13. A pulley system has four strands of rope supporting the load. What is
its mechanical advantage?
14. A screw is very similar to which other type of simple machine?
Section 4.3
15. Why can’t the output work for a machine be greater than the input
work? Explain.
16. Can a simple machine’s efficiency ever be greater than 100%?
Explain your answer.
17. List two examples of ways to increase efficiency in a machine.
Section 4.2
8.
A lever has an input force of 5 newtons and an output force of
15 newtons. What is the mechanical advantage of the lever?
9.
A simple machine has a mechanical advantage of 5. If the output
force is 10 N, what is the input force?
Solving Problems
Section 4.1
1.
2.
3.
4.
5.
Calculate the amount of work you do in each situation.
a. You push a refrigerator with a force of 50 N and it moves 3
meters across the floor.
b. You lift a box weighing 25 N to a height of 2 meters.
c. You apply a 500 N force downward on a chair as you sit on it
while eating dinner.
d. You lift a baby with a mass of 4 kg up 1 meter out of her crib.
e. You climb a mountain that is 1000 meters tall. Your mass is 60
kg.
Sal has a weight of 500 N. How many joules of work has Sal done
against gravity when he reaches 4 meters high on a rock climbing
wall?
You do 200 joules of work against gravity when lifting your backpack
up a flight of stairs that is 4 meters tall. What is the weight of your
backpack in newtons?
You lift a 200 N package to a height of 2 meters in 10 seconds.
a. How much work did you do?
b. What was your power?
One machine can perform 500 joules of work in 20 seconds. Another
machine can produce 200 joules of work in 5 seconds. Which
machine is more powerful?
10. You use a rope and pulley system with a mechanical advantage of 5.
How big an output load can you lift with an input force of 200 N?
11. A lever has an input arm 50 cm long and an output arm 20 cm long.
a. What is the mechanical advantage of the lever?
b. If the input force is 100 N, what is the output force?
12. You want to use a lever to lift a 2000 N rock. The maximum force you
can exert is 500 N. Draw a lever that will allow you to lift the rock.
Label the input force, output force, fulcrum, input arm, and output
arm. Specify measurements for the input and output arms. State the
mechanical advantage of your lever.
13. A rope and pulley system is used so that a 20 N force can lift a 60 N
weight. What is the minimum number of ropes in the system that
must support the weight?
14. A rope and pulley system has two ropes supporting the load.
a. Draw a diagram of the pulley system.
b. What is its mechanical advantage?
c. What is the relationship between the input force and the output
force?
d. How much can you lift with an input force of 20 N?
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15. You push a heavy car weighing 500 newtons up a ramp. At the top of
the ramp, it is 2 meters higher than it was initially.
a. How much work did you do on the car?
b. If your input force on the car was 200 newtons, how long is the
ramp?
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b.
c.
d.
Estimate the power of each light bulb, or get it from the bulb
itself where it is written on the top.
Calculate the total power used by all the bulbs.
Calculate how many horses it would take to make this much
power.
Section 4.3
Section 4.2
16. A lever is used to lift a heavy rock
that weighs 1000 newtons. When a
50-newton force pushes one end of
the lever down 1 meter, how far
does the load rise?
2.
Look for simple machines in your home. List as many as you can
find.
3.
A car is made of a large number of simple machines all working
together. Identify at least five simple machines found in a car.
4.
Exactly how the ancient pyramids of Egypt were built is still a
mystery. Research to find out how simple machines may have been
used to lift the huge rocks of which the pyramids are constructed.
17. A system of pulleys is used to lift
an
elevator
that
weighs
3,000 newtons. The pulley system uses three ropes to support the
load. How far would 12,000 joules of input work lift the elevator?
Assume the pulley system is frictionless.
Section 4.3
5.
Section 4.4
18. A 60 watt light bulb uses 60 joules of electrical energy every second.
However, only 6 joules of electrical energy is converted into light
energy each second.
a. What is the efficiency of the light bulb? Give your answer as a
percentage.
b. What do you think happens to the “lost” energy?
A perpetual motion machine is a machine that, once given energy,
transforms the energy from one form to another and back again
without ever stopping. You have probably seen a Newton’s cradle like
the one shown below.
19. The work output is 300 joules for a machine that is 50% efficient.
What is the work input?
20. A machine is 75% efficient. If 200 joules of work are put into the
machine, how much work output does it produce?
a.
b.
Applying Your Knowledge
c.
Section 4.1
1.
Imagine we had to go back to using horses for power. The power of
one horse is 746 watts (1 horsepower). How many horses would it
take to light up all the light bulbs in your school?
a. First, estimate how many light bulbs are in your school.
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CHAPTER 4: MACHINES, WORK, AND ENERGY
6.
Is a Newton’s cradle a perpetual motion machine?
According to the laws of physics, is it possible to build a
perpetual motion machine?
Many people have claimed to have built perpetual motion
machines in the past. Use the internet to find one such machine.
Explain how it is supposed to work and why it is not truly a
perpetual motion machine.
A food Calorie is equal to 4184 joules. Determine the number of
joules of energy you take in on a typical day.